Evaluate the iterated integral. $ \int_{-2}^0 \left( \int_0^1 xy + x^2 - y^2 \, dy \right) dx =$ Choose 1 answer: Choose 1 answer: (Choice A) A $1$ (Choice B) B $9$ (Choice C) C $-2$ (Choice D) D $5$
Solution: Evaluate the inner integral: $\begin{aligned} \int_{-2}^0 \left( \int_0^1 xy + x^2 - y^2 \, dy \right) dx &= \int_{-2}^0 \left[ \dfrac{xy^2}{2} + x^2y - \dfrac{y^3}{3} \right]_0^1 dx \\ \\ &= \int_{-2}^0 \dfrac{x}{2} + x^2 - \dfrac{1}{3} \, dx \end{aligned}$ Evaluate the outer integral: $\begin{aligned} \int_{-2}^0 \dfrac{x}{2} + x^2 - \dfrac{1}{3} \, dx &= \dfrac{x^2}{4} + \dfrac{x^3}{3} - \dfrac{x}{3} \bigg|_{-2}^0 \\ \\ &= 0 - \left( \dfrac{4}{4} - \dfrac{8}{3} - \dfrac{-2}{3} \right) \\ \\ &= -1 + \dfrac{6}{3} \\ \\ &= 1 \end{aligned}$ The answer: $ \int_{-2}^0 \left( \int_0^1 xy + x^2 - y^2 \, dy \right) dx = 1$